A Radon-Nikodym theorem for finitely additive set functions
Charles Fefferman · Pacific Journal of Mathematics · 1967
Suppose that Σ is a field of subsets of the set S, and suppose that μ and γ are complex-valued finitely additive set functions defined on Σ. Assume that μ is bounded and γ is finite and absolutely continuous with respect to μ. (A word of warning is in order here.The statement "γ is absolutely continuous with respect to μ" is often interpreted as "μ(E) = 0 implies γ(E) = 0".This is not the meaning used here.Our definition is "for every e > 0 there is a δ > 0 such that | μ(E) \ < δ implies I γ(E) I < ε." Unless μ is bounded and countably additive, the two definitions are not equivalent.)